Transport coefficients of hot magnetized QCD matter beyond the lowest Landau level approximation
TTransport coefficients of hot magnetized QCD matter beyond the lowest Landau levelapproximation
Manu Kurian a , ∗ Sukanya Mitra b , † Snigdha Ghosh a , ‡ and Vinod Chandra a § a Indian Institute of Technology Gandhinagar, Gandhinagar-382355, Gujarat, India and b National Superconducting Cyclotron Laboratory,Michigan State University, East Lansing, Michigan 48824, USA
In this article, shear viscosity, bulk viscosity, and thermal conductivity of a hot QCD mediumhave been studied in the presence of strong magnetic field. To model the hot magnetized QCDmatter, an extended quasi-particle description of the hot QCD equation of state in the presenceof the magnetic field has been adopted. The effects of higher Landau levels on the temperaturedependence of viscous coefficients (bulk and shear viscosities) and thermal conductivity have beenobtained by considering the 1 → Keywords : Quark-gluon-plasma, Effective kinetic theory, Strong magnetic field, Thermalrelaxation time, Transport coefficients, Landau levels.
PACS : 12.38.Mh, 13.40.-f, 05.20.Dd, 25.75.-q
I. INTRODUCTION
Relativistic heavy-ion collision (RHIC) experimentshave reported the presence of strongly coupled matter-Quark-gluon plasma (QGP) as a near-ideal fluid [1, 2].The quantitative estimation of the experimental observ-ables such as the collective flow and transverse momen-tum spectra of the produced particles from the hydro-dynamic simulations involve the dependence upon thetransport parameters of the medium. Thus, the trans-port coefficients are the essential input parameters forthe hydrodynamic evolution of the system.Recent investigations show that intense magnetic fieldis created in the early stages of the non-central asymmet-ric collisions [3–6]. This magnetic field affects the ther-modynamic and transport properties of the hot denseQCD matter produced in the RHIC. Ref [7] describesthe extension of ECHO-QGP [8, 9] to the magnetohy-drodynamic regime. The recent major developments re-garding the intense magnetic field in heavy-ion collisioninclude the chiral magnetic effect [10–12], chiral vorticaleffects [13–15] and very recent realization of global Λ-hyperon polarization in non-central RHIC [16, 17]. Thissets the motivation to study the transport coefficients inpresence of the strong magnetic field. The transport pa-rameters under investigation are the viscous coefficients(shear and bulk) and the thermal conductivity of the hot ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] magnetized QGP. Importance of the transport processesin RHIC is well studied [18] and reconfirmed by the re-cent ALICE results [19–21].Quantizing quark/antiquark field in the presence ofstrong magnetic field background gives the Landau lev-els as energy eigenvalues. The quark/antiquark degreesof freedom is governed by (1 + 1) − dimensional Landaulevel kinematics whereas gluonic degrees of freedom re-main intact in the presence of magnetic field [22, 23].However, gluons can be indirectly affected by the mag-netic field through the quark loops while defining theDebye mass of the system.Shear and bulk viscosities can be estimated fromGreen-Kubo formulation both in the presence and ab-sence of magnetic field [22, 24–26]. Lattice results forthe shear and bulk viscosities to entropy ratio are alsowell investigated [27–29]. Viscous pressure tensor quan-tifies the energy-momentum dissipation with the space-time evolution and is characterized by seven viscous co-efficients in the strong magnetic field [30]. The sevenviscous coefficients consist of two bulk viscosities (bothtransverse and longitudinal) and five shear viscosities.The present investigations are focused on the longitudinalcomponent (along the direction of (cid:126)B ) of shear and bulkviscosities since other components of viscosities are negli-gible in the strong magnetic field. Another key transportcoefficient under investigation is the thermal conductiv-ity of the QGP medium. The temperature dependenceof thermal conductivity has been studied in the absenceof magnetic field in the Ref.[31]. The shear and bulkviscosities, electric and thermal conductivities and theirrelative significance have been studied in Ref. [32] withina quasiparticle description of interacting hot QCD equa- a r X i v : . [ nu c l - t h ] D ec tions of state. The first step towards the estimation oftransport coefficients from the effective kinetic theory isto include proper collision integral for the processes inthe strong field. This can be done within the relaxationtime approximation (RTA). Microscopic processes or in-teractions are the inputs of the transport coefficients andare incorporated through thermal relaxation times. Notethat the 1 → viz., the hard thermal loop effective theory(HTL) [35–37] and the relativistic semi-classical trans-port theory [33, 38–41]. The present analysis is donewith the relativistic transport theory by employing theChapman-Enskog method. Hot QCD medium effects areencoded in the quark/antiquark and gluonic degrees offreedom by adopting the effective fugacity quasiparticlemodel (EQPM) [23, 42–44]. The transport coefficientspick up the mean field term (force term) as describedin Ref [45]. The mean field term comes from the lo-cal conservations of number current and stress-energytensor in the covariant effective kinetic theory. In thecurrent analysis, we investigate the mean field correc-tions in the presence of strong magnetic field and studythe temperature behaviour of the transport coefficients.Here, the strong magnetic field restricts the calculationsto (1+1) − dimensional (dimensional reduction) covarianteffective kinetic theory for quarks and antiquarks.The manuscript is organized as follows. In section II,the mathematical formulation for the estimation of trans-port coefficients from the effective covariant kinetic the-ory is discussed along with the quasiparticle descriptionof hot QCD medium in the strong magnetic field. Sec-tion III deals with the thermal relaxation for the 1 → II. FORMALISM: TRANSPORT COEFFICIENTSAT STRONG MAGNETIC FIELD
The strong magnetic field (cid:126)B = B ˆ z constraints thequarks/antiquarks motion parallel to field with a trans-verse density of states. The viscous coefficients [22, 46]and heavy quark diffusion coefficient [47] have been per-turbatively calculated under the regime α s | q f eB |(cid:28) T (cid:28)| q f eB | with the lowest Landau level (LLL) ap-proximation. But the validity of LLL approximation isquestionable since higher Landau level contributions aresignificant at | eB | = 10 m π in the temperature range above 200 MeV. Here, we are focusing on the more re-alistic regime gT (cid:28) (cid:112) | q f eB | in which higher Lan-dau level (HLL) contributions are significant. In thevery recent work [33], Fukushima and Hidaka have beenestimated the longitudinal conductivity of magnetizedQGP with full Landau level resummation in the regime gT (cid:28) (cid:112) | q f eB | .The formalism for the estimation of transport coef-ficients includes the quasiparticle modeling of the sys-tem away from the equilibrium followed by the set-ting up of the effective kinetic theory for different pro-cesses. Quasiparticle models encode the medium ef-fects, viz. , effective fugacity or with effective mass. Thelater include self-consistent and single parameter quasi-particle models [48], NJL and PNJL based quasiparti-cle models [49], effective mass with Polyakov loop [50]and recently proposed quasiparticle models based onthe Gribov-Zwanziger (GZ) quantization [51–53]. Here,the analysis is done within the effective fugacity quasi-particle model (EQPM) where the medium interactionsare encoded through temperature dependent effectivequasigluon and quasiquark/antiquark fugacities, z g and z q respectively. The extended EQPM describes the hotQCD medium effects in strong magnetic field [23]. Weconsidered the (2+1) flavor lattice QCD equation of state(EoS) (LEoS) [54, 55] and the 3-loop HTLpt EOS [56, 57]for the effective description of QGP in strong magneticfield [23, 46]. Transport coefficients from effective (1+1)-D kinetictheory
In the absence of magnetic field, the particle four flow¯ N µ ( x ) can be defined in terms of quasiparticle (dressed)momenta (cid:126) ¯ p k within EQPM as [45],¯ N µ ( x ) = N (cid:88) k =1 ν k (cid:90) d | (cid:126) ¯ p k | (2 π ) ω k ¯ p µk f k ( x, ¯ p k )+ N (cid:88) k =1 δων k (cid:90) d | (cid:126) ¯ p k | (2 π ) ω k (cid:104) ¯ p µk (cid:105) E k f k ( x, ¯ p k ) , (1)in which ν k is the degeneracy factor of the k th species.Here, we are considering non-zero masses ( m f ) for quarks(up, down and strange quarks with masses m u = 3 MeV, m d = 5 MeV and m s = 100 MeV respectively) and hence E k = (cid:113) | (cid:126) ¯ p k | + m f for quarks/antiquarks and E k = | (cid:126) ¯ p | for gluons. The term (cid:104) ¯ p µ (cid:105) = ∆ µν ¯ p ν is the irreducibletensor with ∆ µν = g µν − u µ u ν as the projection operator.The metric has the form g µν =diag (1 , − , − , − u µ ≡ (1 , ) is given by, f q,g = z q,g exp [ − β ( u µ p µ )]1 ± z q,g exp [ − β ( u µ p µ )] , (2)with p µ = ( E, (cid:126) ¯ p ). Quasiparticle momenta (dressed mo-menta) and bare particle four-momenta can be relatedfrom the dispersion relations as,¯ p µ = p µ + δωu µ , δω = T ∂ T ln( z q,g ) , (3)which modifies the zeroth component of the four-momenta in the local rest frame. Hence, we have¯ p ≡ ω k = E k + δω. (4)The dispersion relation in Eq. (4) encodes the collectiveexcitation of quasiparton along with the single particleenergy. Also, the energy-momentum tensor ¯ T µν in termsof dressed momenta takes the following form,¯ T µν ( x ) = N (cid:88) k =1 ν k (cid:90) d | (cid:126) ¯ p k | (2 π ) ω k ¯ p µk ¯ p νk f k ( x, ¯ p k )+ N (cid:88) k =1 δων k (cid:90) d | (cid:126) ¯ p k | (2 π ) ω k (cid:104) ¯ p µk ¯ p νk (cid:105) E k f k ( x, ¯ p k ) , (5)where (cid:104) ¯ p µk ¯ p νk (cid:105) = 12 (∆ µα ∆ νβ + ∆ µβ ∆ να )¯ p α ¯ p β .In our case, Eq. (5) should rewritten for the hot QCDmedium in the strong magnetic field (cid:126)B = B ˆ z limit.Thereafter, the transport coefficients could be obtainedby realizing the microscopic (transport theory) definitionof ¯ T µν to the macroscopic decomposition at various or-der. Recall that the EQPM in the presence of a strongmagnetic field is studied by considering the Landau leveldynamics in the dispersion relation for quarks whereasgluonic part remain invariant in magnetic field [23, 46].The quasi-quark/antiquark distribution function in thestrong magnetic field background takes the form as inEq. (2) with the particle four-momenta p µ (cid:107) = ( ω l , , , ¯ p z ).The zeroth component of four-momenta becomes,¯ p ≡ ω l = (cid:113) ¯ p z + m f + 2 l | q f eB | + δω. (6)where (cid:113) ¯ p z + m f + 2 l | q f eB | ≡ E l is the Landau levelenergy eigenvalue in the strong magnetic field.Macroscopically, the energy-momentum tensor in thepresence of magnetic field (cid:126)B = B ˆ z can be decomposedas [22], ¯ T µν = εu µ u ν − P ⊥ Ξ µν + P (cid:107) b µ b ν + τ µν , (7)where u µ is the flow vector and b µ = (cid:15) µναβ F να u β / (2 B )with B = (cid:112) − B µ B µ . Here, P ⊥ and P (cid:107) are the transverseand longitudinal components of pressure respectively andholds the relation P ⊥ = P (cid:107) − M B , where the magnetiza-tion M = ( ∂P∂B ) T . The tensor Ξ µν = ∆ µν + b µ b ν , projectsout the two-dimensional space orthogonal to both theflow and magnetic field. In the presence of strong mag-netic field, the pressure can be defined as, P = P (cid:107) q + P g , (8)with P ≡ ¯ T µν b µ b ν = ¯ T . Here, P (cid:107) q is the dominantquark and antiquark contribution to the pressure in the strong magnetic field [22, 23] and have the following form, P (cid:107) = ∞ (cid:88) l =0 | q f eB | π N c (cid:90) ∞ dp z p z E l µ l f q . (9)The integration phase factor in the strong field due todimensional reduction [58–60] is defined as, (cid:90) d p (2 π ) → | q f eB | π (cid:90) ∞−∞ dp z π µ l , (10)where µ l = (2 − δ l ) is the spin degeneracy factor of theLandau levels. Since gluonic dynamics are not directlyaffected by the magnetic field, the gluonic contribution P g retains the same form as in the absence of magneticfield and is well investigated in the work [42]. Note that inthe presence of the strong magnetic field quark/antiquarkcontribution is dominant compared with that of glu-ons [22, 33, 34]. Also, we can define the quark and anti-quark contribution to energy density in the strong fieldas, ε (cid:107) = ∞ (cid:88) l =0 | q f eB | π N c (cid:90) ∞ dp z ( ω p ) ω l µ l f q . (11)Since the quark dynamics is constrained in the (1 + 1)-dimensional space, both b µ and u µ are longitudinal(1 + 1)-dimensional vector and at the same time b µ isorthogonal to u µ . The longitudinal projection operator∆ µν (cid:107) is perpendicular to u µ and can constructed from b µ [61] as, ∆ µν (cid:107) ≡ g µν (cid:107) − u µ u ν = − b µ b ν , (12)where g µν (cid:107) = diag (1 , , , − T µν = ε (cid:107) u µ u ν − P (cid:107) ∆ µν (cid:107) . (13)In the strong magnetic field, T µν can be defined in termsof quasiparticle momenta of quarks and antiquarks as thefollowing, T µν ( x ) = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ¯ p (cid:107) µk ¯ p (cid:107) νk × f k ( x, ¯ p z k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105) E l k × f k ( x, ¯ p z k ) , (14)which give back the expressions as in Eqs. (9) and (11)for the pressure and energy density respectively throughthe following definitions, ε (cid:107) = u µ u ν T µν , P (cid:107) = ∆ µν (cid:107) T µν . (15)Here, ¯ p (cid:107) µk ≡ ( ω l k , , , p z k ) incorporates the longi-tudinal components and (cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105) = 12 (∆ µα (cid:107) ∆ νβ (cid:107) +∆ µβ (cid:107) ∆ να (cid:107) ) ¯ p (cid:107) α ¯ p (cid:107) β . For the weak (moderate) magneticfield, one also needs to analyse the transverse dynamics ofthe hot QCD matter. In these situations, the transversecomponents of various transport coefficients might playa significant role. These aspects are beyond the scope ofthe present work and the matter of future extensions ofthe work. Following the above arguments, four flow N µ of the quarks and antiquarks in the strong magnetic fieldhas the following form, N µ ( x ) = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ¯ p (cid:107) µk × f k ( x, ¯ p z k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104) ¯ p (cid:107) µk (cid:105) E l k × f k ( x, ¯ p k ) , (16)with (cid:104) ¯ p (cid:107) µ (cid:105) = ∆ (cid:107) µν ¯ p (cid:107) ν .Estimation of the transport coefficients requires thesystem away from equilibrium. In the current analysis,we are focusing on the dominant quark/antiquark dy-namics of the magnetized QGP. Here, we need to set-upthe relativistic transport equation, which quantifies therate of change of quasiquark/antiquark distribution func-tion in terms of collision integral. The thermal relaxationtime ( τ eff ) linearize the collision term ( C ( f q )) in the fol-lowing way,1 ω l k ¯ p (cid:107) µk ∂ µ f k ( x, ¯ p z k ) + F z ∂f k ∂p z k = C ( f k ) ≡ − δf k τ eff , (17)with F z = − ∂ µ ( δωu µ u z ) is the force term from the con-servation of particle density and energy momentum [45].The local momentum distribution function of quarks canexpand as, f k = f k ( p z ) + δf k , δf k = f k (1 ± f k ) φ k . (18)Here, φ k defines the deviation of the quasiquark distri-bution function from its equilibrium. The Eq. (17) givesthe effective kinetic theory description of the quasipar-tons under EQPM in the strong magnetic field. In or-der to estimate the transport coefficients, we employ theChapman-Enskog (CE) method. Applying the definitionof equilibrium quasiparton momentum distribution func-tion as in Eq. (2), the first term of Eq. (17) gives thenumber of terms with thermodynamic forces of the trans-port processes. The second term of Eq. (17) vanishes fora co-moving frame. Finally, we are left with, Q k X + (cid:104) ¯ p (cid:107) µk (cid:105) ( ω l k − h k ) X qµ −(cid:104)(cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105)(cid:105) X µν = − T ω p k τ eff φ k , (19)in which the conformal factor due to the dimensional re-duction in the strong field limit is Q k = (¯ p z k − ω l k c s ) where c s is the speed of sound and h k is the enthalpyper particle of the system that can be defined fromthe basic QCD thermodynamics. Here, (cid:104)(cid:104) P µ (cid:107) R ν (cid:107) (cid:105)(cid:105) = (cid:110) ∆ (cid:107) µα ∆ (cid:107) νβ + ∆ (cid:107) µβ ∆ (cid:107) να − ∆ (cid:107) αβ ∆ µν (cid:107) (cid:111) P α (cid:107) R β (cid:107) . Thebulk viscous force, thermal force and shear viscous forceare defined respectively as follows, X = ∂.u, (20) X µq = (cid:26) (cid:53) µ TT − (cid:53) µ Pnh (cid:27) , (21) X µν = (cid:104)(cid:104) ∂ µ u ν (cid:105)(cid:105) , (22)where h is the total enthalpy defined as h = (cid:80) Nk =0 h k and n is the total number density of the system. Note thathere µ = 0 , φ k that is the linear combination of these forces can berepresented as, φ k = A k X + B µk X qµ − C µνk X µν , (23)where the coefficients can be defined from Eq. (19) as, A k = Q k {− T ω lk τ eff } , (24) B µk = (cid:104) ¯ p µk (cid:105) ( ω l k − h k ) {− T ω lk τ eff } , (25) C µνk = (cid:104)(cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105)(cid:105){− T ω lk τ eff } . (26)Following this formalism, we can estimate the viscous co-efficients and thermal conductivity of the QGP mediumin the strong magnetic field.
1. Shear and bulk viscosity
We can define the pressure tensor from the energy-momentum tensor as in the following way, P µν = ∆ (cid:107) µσ T στ ∆ (cid:107) ντ . (27)We can decompose the P µν in equilibrium and non-equilibrium components of distribution function as fol-lows, P µν = − P ∆ (cid:107) µν + Π µν , (28)where Π µν is the viscous pressure tensor. Following thedefinition of T µν as in Eq. (14), Π µν takes the form,Π µν = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105)× δf k ( x, ¯ p z k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105) E l k × δf k ( x, ¯ p z k ) . (29)In the very strong magnetic field, the pressure tensor hasdifferent form as compared to the case without magneticfield. This is due to the (1+1) − dimensional energy eigen-values of the quarks and antiquarks. Hence, µ and ν canbe 0 or 3 in the strong magnetic field, describing the lon-gitudinal components of the viscous pressure tensor. Theform of viscous pressure tensor in the strong magneticfield is described in the recent works by Tuchin [30, 62].Magnetized plasma is characterized by five shear compo-nents. Among the five coefficients, four components arenegligible when the strength of the magnetic field is suf-ficiently higher than the square of the temperature [63].Here, we are focusing on the non-negligible longitudinalcomponent of shear and bulk viscous coefficients of thehot QGP medium in the strong magnetic field.Following [32], the longitudinal shear viscous tensorhas the following form,¯Π µν = Π µν − Π∆ (cid:107) µν = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104)(cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105)(cid:105)× f k (1 − f k ) φ k + ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104)(cid:104) ¯ p (cid:107) µk ¯ p (cid:107) νk (cid:105)(cid:105) E l k × f k (1 − f k ) φ k . (30)Also, the bulk viscous part in the longitudinal directioncomes out to be,Π = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ∆ (cid:107) µν × ¯ p (cid:107) µk ¯ p (cid:107) νk f k (1 − f k ) φ k + ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ∆ (cid:107) µν × ¯ p (cid:107) µk ¯ p (cid:107) νk E l k f k (1 − f k ) φ k . (31)Substituting φ k from Eq. (23) and comparing with themacroscopic definition Π µν = 2 η (cid:104)(cid:104) ∂ µ u ν (cid:105)(cid:105) + ζ ∆ (cid:107) µν ∂.u , wecan obtain the expressions of longitudinal viscosity coef-ficients in the strong field limit. Note that the longitudi-nal component of shear viscosity, i.e., in the direction ofmagnetic field, is defined from ¯Π [63]. The longitudinalshear η and bulk viscosity ζ are obtained as, η = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) | ¯ p z k | ω l k τ eff f k (1 − f k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) 1 ω l k | ¯ p z k | E l k τ eff × f k (1 − f k ) , (32) and ζ = ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) 1 ω l k { ¯ p z k − ω l k c s } × τ eff f k (1 − f k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) 1 ω l k { ¯ p z k − ω l k c s } × E l k τ eff f k (1 − f k ) . (33)The second term in the Eq. (32) and Eq. (33) gives cor-rection to viscous coefficients due to the quasiparton ex-citations whereas the first term comes from the usualkinetic theory of bare particles.
2. Thermal conductivity
The heat flow is the difference between the energy flowand enthalpy flow by the particle, I µq = u ν T νσ ∆ (cid:107) µσ − hN σ ∆ (cid:107) µσ . (34)In terms of the modified/non-equilibrium distributionfunction Eq. (34) becomes, I µ = u ν ∆ (cid:107) µσ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ¯ p (cid:107) νk ¯ p (cid:107) σk δf k − h ∆ (cid:107) µσ (cid:34) ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k ¯ p (cid:107) σk δf k ( x, ¯ p z k )+ ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q δωµ l | q f k eB | π N c (cid:90) ∞−∞ d ¯ p z k (2 π ) ω l k (cid:104) ¯ p (cid:107) σk (cid:105) E l k δf k ( x, ¯ p z k ) (cid:35) , (35)in which heat flow retains only non-equilibrium part ofthe distribution function. After contracting with projec-tion operator and hydrodynamic velocity along with thesubstitution of δf k from Eq. (17) and comparing with themacroscopic definition of heat flow, we obtain I µ = λT X µq . (36)We obtain the thermal conductivity in the strong mag-netic field as, λ = (cid:40) ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) τ eff ( ω l k − h k ) ω l k × | ¯ p z k | f k (1 − f k ) (cid:41) − (cid:40) ∞ (cid:88) l =0 (cid:88) k ∈ q, ¯ q µ l δω | q f k eB | π N c T (cid:90) ∞−∞ d ¯ p z k (2 π ) τ eff h k ( ω l k − h k ) ω l k × | ¯ p z k | E l k f k (1 − f k ) (cid:41) . (37)The second term with δω in the heat flow comes fromthe N µ which encodes the quasiparticle excitation in thethermal conductivity. III. THERMAL RELAXATION IN THESTRONG MAGNETIC FIELD
Thermal relaxation is the essential dynamical input ofthe transport processes which counts for the microscopicinteraction of the system. In the strong magnetic field,the 1 → → τ eff ,can be defined from the relativistic transport equationin terms of distribution function in the strong magneticfield (cid:126)B = B ˆ z as, df q dt = C ( f q ) ≡ − δf q τ eff . (38)Here, C ( f q ) represents the collision integral for the pro-cess under consideration. For the 1 → p + p (cid:48) −→ k , where primed notation for antiquark), thethermal relaxation in the strong magnetic field can bedefined as follows, τ − ( p z ) = ∞ (cid:88) l (cid:48) =0 (cid:90) ∞−∞ dp (cid:48) z π (cid:90) d k (2 π ) (2 π ) δ ( k z − p z − p (cid:48) z )2 ω k ω l p ω l p (cid:48) × | M p + p (cid:48) → k | f q ( p (cid:48) z )(1 + f g ( k ))(1 − f q ( p z )) , (39)where the quasiquark distribution function is defined as, f q = z q exp ( − β (cid:113) p z + m f + 2 l | q f eB | )1 + z q exp ( − β (cid:113) p z + m f + 2 l | q f eB | ) , (40)and the quasigluon distribution function has the form, f g = z g exp ( − β | (cid:126)k | )1 − z g exp ( − β | (cid:126)k | ) . (41)Within the LLL approximation the momentum depen-dent thermal relaxation time takes the following form inthe regime p z (cid:48) ∼
0, as [46, 64],( τ − ) l =0 = 2 α eff C F m f ω q (1 − f q ) z q ( z q + 1) (1 + f g ( E p z )) ln ( T /m ) , (42)where C F is the Casimir factor of the processes and α eff is the effective coupling constant defined from the Debyescreening mass [46].The Impact of the higher Landau levels on the matrixelement and distribution function for the 1 → → τ − ( p z ) = 14 ω l q − f q ( p z )) ∞ (cid:88) l (cid:48) ≥ l (cid:90) ∞−∞ dp (cid:48) z π ω l (cid:48) ¯ q X ( l, l (cid:48) , ξ ) × f q ( p (cid:48) z )(1 + f g ( p (cid:48) z + p z )) , (43)where ξ is defined as, ξ = ( ω l q + ω l (cid:48) ¯ q ) − ( p z + p (cid:48) z ) | q f eB | , (44)and X ( l, l (cid:48) , ξ ) takes the form as follows, X ( l, l (cid:48) , ξ ) = 4 πα eff N c C F l ! l (cid:48) ! e − ξ ξ l (cid:48) − l (cid:34)(cid:18) m f − | q f eB | ( l + l (cid:48) − ξ ) 1 ξ ( l + l (cid:48) ) (cid:19) F ( l, l (cid:48) , ξ )+ 16 | q f eB | l (cid:48) ( l + l (cid:48) ) 1 ξ L ( l (cid:48) − l ) l ( ξ ) L ( l (cid:48) − l ) l − ( ξ ) (cid:35) , (45)with F ( l, l (cid:48) , ξ ) = [ L ( l (cid:48) − l ) l ( ξ )] + l (cid:48) l [ L ( l (cid:48) − l ) l − ( ξ )] for l > π , Upto 50 LL < τ > (f m / c ) T/T c updownstrange 0 3 6 9 12 15 18 1.5 2 2.5 3 3.5 4 FIG. 1: The temperature dependence of thermalrelaxation time for quarks at | eB | = 10 m π .and F ( l, l (cid:48) , ξ ) = 1 for the lowest Landau level. Here, α eff is the effective coupling constant and is defined from theDebye screening masses of the QGP [32, 65–68].Hot medium effects are entering through the quasi-parton distribution function and the effective coupling.The effective thermal relaxation time controls the be-haviour of transport coefficients critically. Note that inthe limit T (cid:28)| q f eB | , LLL approximation is validso that X ( l = 0 , l (cid:48) = 0 , ξ ) ≈ π ( α eff ) m f N c C F , where π ζ / s T/T c Lowest LLUpto 5 LLUpto 20 LLUpto 50 LLB=0, Mitra et al. 1 10 100 1.5 2 2.5 3 3.5 4
FIG. 2: The effects of HLLs on the temperaturebehaviour of ζ/s at | eB | = 10 m π . Behaviour of ζ/s iscomparing with the result at B = 0 of Mitra et al. [45]. π η / s T/T c Lowest LLUpto 5 LLUpto 20 LLUpto 50 LLB=0, LatticeB=0, Marty et al. 0.1 1 10 100 1.5 2 2.5 3 3.5 4
FIG. 3: The effects of HLLs on the temperaturebehaviour of η/s at | eB | = 10 m π . Lattice data [27] andresult of Marty et al. [31] for η/s are in the absence ofmagnetic field. e − ξ ≈ T (cid:28)| q f eB | .Following the parton distribution function within theEQPM framework, the thermal average of τ eff can bedefined as, < τ eff > = (cid:80) ∞ l =0 (cid:82) ∞−∞ dp z τ eff f q (cid:80) ∞ l =0 (cid:82) ∞−∞ dp z f q . (46)Notably, the thermal average is taken merely to explorethe temperature behaviour of < τ eff > with the inclusion of the effects of HLLs and analysed in the next section.While computing the transport coefficients the momen-tum dependence of the relaxation time, τ eff has beenemployed. IV. RESULTS AND DISCUSSIONS
Let us initiate the discussion with the temperaturebehaviour of thermal relaxation time τ eff of the quarks(up, down and strange quarks with masses m u = 3 MeV, m d = 5 MeV and m s = 100 MeV respectively) for thedominant 1 → TT c for | eB | = 10 m π considering upto 50 LLs in the Fig. 1. The relaxation time exhibitsthe decreasing trend with increasing temperature. In thelimit, T (cid:28)| q f eB | , τ eff defined in Eq. (43) reduced tothe LLL result as described in [46]. To encode the EoSeffects in the thermal relaxation, the quasiparticle par-ton distribution functions are introduced along with theeffective coupling constant. The thermal relaxation timeact as the dynamical input for the transport processes.Following the Eq. (33), the temperature dependenceof bulk viscosity depends on the term ω p ( p z k − ω p c s ) and the relaxation time τ eff , where c s can be obtainedfrom the QCD thermodynamics. The ratio of longitu-dinal bulk viscosity to entropy density for the 1 → | eB | = 10 m π has been plotted as a functionof T /T c in the Fig. 2. The temperature dependence of the ζ/s in the strong magnetic field indicates its rising be-haviour near T c . The behaviour of longitudinal shear vis-cosity for the 1 → T /T c at | eB | = 10 m π is shown in Fig. 3. Since the driving force for the longi-tudinal shear viscosity is in the direction of the magneticfield, the Lorentz force does not interfere in the calcu-lation. Quantitatively, η/s with the HLL contributionsremains within the same range of the lattice data [27]and NJL model result in [31] at B = 0. This observationis in line with the result that longitudinal conductivitywith HLLs contributions remains within the range of thelattice result at zero magnetic field [33]. For the numer-ical estimation of ζ/s and η/s , we truncate the Landaulevel sum at l max = 50. We observe that the HLL con-tributions are significant in the estimation of the viscouscoefficients whereas the LLL approximation has an en-hancement as m f tends to zero. Our observations on theeffects of HLLs to the transport coefficients are qualita-tively consistent with the results of the recent work ofFukushima and Hidaka [33].The present analysis is done by employing the effec-tive covariant kinetic theory using the Chapman-Enskogmethod including the effects of HLLs. The mean fieldforce term which emerges from the effective theory in-deed appears as the mean field corrections to the trans-port coefficients. The second term in the Eq. (32) andEq. (33) describes the mean field contribution to the lon-gitudinal shear viscosity and bulk viscosity in the strong π , Upto 50 LLT/T c η /s - With Mean Field η /s - Without Mean Field ζ /s - With Mean Field ζ /s - Without Mean Field 0.4 0.6 0.8 1 1.2 1.4 1.5 2 2.5 3 3.5 4 η /s ζ /s 0.8 1.2 1.6 2 1 2 3 4 5 6 FIG. 4: Temperature dependence of ζ/s and η/s with and without mean field correction at | eB | = 10 m π for 1 → ζ/s and η/s (right panel).
10 100 1000 1.5 2 2.5 3 3.5 4eB = 10 m π λ / T T/T c Marty et al. , B=0Lowest LLUpto 5 LLUpto 20 LLUpto 50 LL 10 100 1000 1.5 2 2.5 3 3.5 4
FIG. 5: Thermal conductivity as a function of
T /T c at | eB | = 10 m π . Behaviour of λ/T is comparing with theresult at B = 0 of Marty et al. [31]magnetic field, respectively. The mean field term con-sists of the term δω which is the temperature gradient ofthe effective fugacity z g/q . The temperature behavioursof the viscous coefficients (bulk and shear viscosities) inthe presence of strong magnetic field with and withoutthe mean field corrections are shown in Fig. 4 (left panel).At higher temperature, the effects are negligible since theeffective fugacity behaves as a slowly varying function oftemperature there. Hence, the mean field corrections dueto the quasiparticle excitations are significant at temper-ature region closer to T c . The magnetic field dependenceof the bulk viscosity and shear viscosity have been plottedin the Fig. 4(right panel). In the strong magnetic field limit, the viscous coefficients could be computed withinLLL approximation. The inclusion of HLLs reflects thenon-trivial (non-monotonic) magnetic field dependence ofthe transport coefficients. Similar non-monotonic struc-ture in the magnetic field dependence of longitudinal con-ductivity with HLLs is described in [33]. The estimationof electric conductivity within our model while includingthe HLLs is beyond the scope of the present analysis andis a matter of future investigations.Mean field corrections to the thermal conductivity isexplicitly shown in Eq. (37) in which thermal relaxationincorporates the microscopic interactions. We depictedthe temperature behaviour of λ/T in Fig. 5. The HLLeffects of the transport coefficients are entering throughthe thermal relaxation time and the quasiparticle dis-tribution function. These effects are significant in theestimation of transport coefficients in the presence of amagnetic field. The temperature behaviour of the dimen-sionless quantity λ/T in the absence of the magneticfield is well investigated [31, 32] and is in the order of100 −
25 within the temperature range (1 − TT c , whichis quantitatively consistent with our result. V. CONCLUSION AND OUTLOOK
In conclusion, we have computed the temperature be-haviour of the transport parameters such as longitudi-nal viscous coefficients (shear and bulk viscosities) andthermal conductivity for the 1 → gT (cid:28) (cid:112) | q f eB | in which higher Landau level (HLL)contributions are significant. Notably, the inclusion ofHLL contributions are essential to explain the transportprocesses at high temperature in the presence of the mag-netic field. Furthermore, effects of the mean field termare seen to be quite significant as fas as the temperaturebehavior of the above mentioned transport coefficients isconcerned (for the temperatures which are not very faraway from T c ).An immediate future extension of the work is to inves-tigate the aspects of non-linear electromagnetic responsesof the hot QGP with the mean field contribution alongwith the effective description of magnetohydrodynamicwaves in the hot QGP medium. In addition, the es- timation of all transport coefficients from covariant ki-netic theory within the effective fugacity quasiparticlemodel using more realistic collision integral, for example,BGK (Bhatnagar, Gross and Krook) collision term, inthe strong magnetic field would be another direction towork. ACKNOWLEDGMENTS
V.C. would like to acknowledge Science and Engineer-ing Research Board (SERB), Govt. of India for theEarly Career Research Award (ECRA/2016) and De-partment of Science and Technology (DST), Govt. ofIndia for INSPIRE-Faculty Fellowship (IFA-13/PH-55).S.G. would to like acknowledge the Indian Institute ofTechnology Gandhinagar for the postdoctoral fellowship.S.M. would like to acknowledge SERB-INDO US forumto conduct the Postdoctoral research in USA. We recordour gratitude to the people of India for their generoussupport for the research in basic sciences. [1] Adams et al. (STAR Collaboration), Nucl. Phys. A ,102 (2005); K. Adcox et al. (PHENIX Collaboration),Nucl. Phys. A , 184 (2005); B.B. Back et al. (PHO-BOS Collaboration), Nucl. Phys. A , 28 (2005); A.Arsence et al. (BRAHMS Collaboration), Nucl. Phys.A , 1 (2005).[2] K. Aamodt et al. [ALICE Collaboration], Phys. Rev.Lett. , 252301 (2010).[3] V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod.Phys. A , 5925 (2009).[4] Y. Zhong, C. B. Yang, X. Cai and S. Q. Feng, Adv. HighEnergy Phys. (2014) 193039.[5] W.-T. Deng, and Xu-Guang Huang, Phys. C , 044907(2012).[6] S. K. Das, S. Plumari, S. Chatterjee, J. Alam, F. Scardinaand V. Greco, Phys. Lett. B , 260 (2017).[7] G. Inghirami, L. Del Zanna, A. Beraudo, M. H. Moghad-dam, F. Becattini and M. Bleicher, Eur. Phys. J. C ,no. 12, 659 (2016).[8] L. Del Zanna et al. , Eur. Phys. J. C , 2524 (2013).[9] F. Becattini et al. , Eur. Phys. J. C , no. 9, 406 (2015).[10] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys.Rev. D , 074033 (2008), D. E. Kharzeev, L. D. McLer-ran, H. J. Warringa, Nucl. Phys. A , 227 (2008),D. E.Kharzeev, Annals Phys. , 205 (2010), D. E. Kharzeevand D. T. Son, Phys. Rev. Lett. 106, 062301 (2011).[11] A. V. Sadofyev and M. V. Isachenkov, Phys. Lett. B , 404 (2011), A. V. Sadofyev, V. I. Shevchenko andV. I. Zakharov, Phys. Rev. D , 105025 (2011).[12] X. G. Huang, Y. Yin, and J. Liao Nucl. Phys. A , 661(2016).[13] D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang,Prog. Part. Nucl. Phys. , 1 (2016). [14] A. Avkhadiev and A. V. Sadofyev, Phys. Rev. D , no.4, 045015 (2017).[15] N. Yamamoto, Phys. Rev. D , no. 5, 051902 (2017).[16] L. Adamczyk et al. [STAR Collaboration], Nature ,62 (2017).[17] F. Becattini, I. Karpenko, M. Lisa, I. Upsal andS. Voloshin, Phys. Rev. C , no. 5, 054902 (2017).[18] M. Luzum and P. Romatschke, Phys. Rev. C , 034915(2008) Erratum: [Phys. Rev. C , 039903 (2009)].[19] J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett. , 182301 (2016), J. Adam et al. [ALICE Collabora-tion], Phys. Rev. Lett. , no. 13, 132302 (2016).[20] B. Abelev et al. [ALICE Collaboration], Phys. Rev. Lett. , no. 23, 232302 (2013).[21] J. Adam et al. [ALICE Collaboration], JHEP , 164(2016).[22] K. Hattori, X. G. Huang, D. H. Rischke and D. Satow,Phys. Rev. D , no. 9, 094009 (2017).[23] M. Kurian and V. Chandra, Phys. Rev. D , no. 11,114026 (2017).[24] D. Kharzeev and K. Tuchin, JHEP , 093 (2008).[25] G. D. Moore and O. Saremi, JHEP , 015 (2008).[26] A. Czajka and S. Jeon, Phys. Rev. C , no. 6, 064906(2017).[27] A. Nakamura and S. Sakai, Phys. Rev. Lett. , 072305(2005).[28] N. Astrakhantsev, V. Braguta and A. Kotov, JHEP , 101 (2017).[29] N. Y. Astrakhantsev, V. V. Braguta and A. Y. Kotov,arXiv:1804.02382 [hep-lat].[30] K. Tuchin, J. Phys. G , 025010 (2012).[31] R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin andH. Berrehrah, Phys. Rev. C , 045204 (2013). [32] S. Mitra and V. Chandra, Phys. Rev. D , no. 9, 094003(2017).[33] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. , no.16, 162301 (2018).[34] K. Hattori, S. Li, D. Satow and H. U. Yee, Phys. Rev. D , no. 7, 076008 (2017).[35] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP ,051 (2003).[36] M. A. Valle Basagoiti, Phys. Rev. D , 045005 (2002).[37] G. D. Moore, JHEP , 039 (2001).[38] J. W. Chen, H. Dong, K. Ohnishi and Q. Wang, Phys.Lett. B , 277 (2010).[39] A. S. Khvorostukhin, V. D. Toneev and D. N. Voskresen-sky, Phys. Rev. C , 035204 (2011).[40] Z. Xu and C. Greiner, Phys. Rev. Lett. , 172301(2008).[41] L. Thakur, P. K. Srivastava, G. P. Kadam, M. Georgeand H. Mishra, Phys. Rev. D , no. 9, 096009 (2017).[42] V. Chandra and V. Ravishankar, Phys. Rev. D ,074013 (2011).[43] V. Chandra, R. Kumar and V. Ravishankar, Phys. Rev.C , 054909 (2007).[44] S. Mitra and V. Chandra, Phys. Rev. D , no. 3, 034025(2016).[45] S. Mitra and V. Chandra, Phys. Rev. D , no. 3, 034032(2018).[46] M. Kurian and V. Chandra, Phys. Rev. D , no. 11,116008 (2018).[47] K. Fukushima, K. Hattori, H. U. Yee and Y. Yin, Phys.Rev. D , no. 7, 074028 (2016)[48] V. M. Bannur, Phys. Rev. C , 044905 (2007); Phys.Lett. B , 271 (2007); JHEP , 046 (2007).[49] A. Dumitru, R. D. Pisarski, Phys. Lett. B , 95 (2002);K. Fukushima, Phys. Lett. B , 277 (2004); S. K.Ghosh et. al, Phys. Rev. D , 114007 (2006); H. Abuki,K. Fukushima, Phys. Lett. B , 57 (2006); H. M. Tsai,B. Muller, J. Phys. G , 075101 (2009).[50] M. D’Elia, A. Di Giacomo and E. Meggiolaro, Phys. Lett.B , 315 (1997); Phys. Rev. D , 114504 (2003); P. Castorina, M. Mannarelli, Phys. Rev. C , 054901(2007); Phys. Lett. B , 336 (2007).[51] N. Su and K. Tywoniuk, Phys. Rev. Lett. , no. 16,161601 (2015).[52] W. Florkowski, R. Ryblewski, N. Su, and K. Tywoniuk,Phys. Rev. C 94 , 044904 (2016); Acta Phys. Pol. B ,1833 (2016).[53] A. Bandyopadhyay, N. Haque, M. G. Mustafa andM. Strickland, Phys. Rev. D , no. 6, 065004 (2016).[54] M. Cheng et al. , Phys. Rev. D , 014511 (2008).[55] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,Kalman K. Szabo, Phys. Lett. B , 99-104 (2014).[56] N. Haque, A. Bandyopadhyay, J. O. Andersen, MunshiG. Mustafa, M. Strickland and Nan Su, JHEP , 027(2014).[57] J. O. Andersen, N. Haque, M. G. Mustafa and M. Strick-land, Phys. Rev. D , 054045 (2016).[58] F. Bruckmann, G. Endrodi, M. Giordano, S. D. Katz,T. G. Kovacs, F. Pittler and J. Wellnhofer, Phys. Rev. D , no. 7, 074506 (2017).[59] A. N. Tawfik, J. Phys. Conf. Ser. , no. 1, 012082(2016).[60] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl.Phys. B , 249 (1996).[61] S. Li and H. U. Yee, Phys. Rev. D , no. 5, 056024(2018).[62] K. Tuchin, Adv. High Energy Phys. , 490495 (2013).[63] D. D. Ofengeim and D. G. Yakovlev, EPL , no. 5,59001 (2015).[64] K. Hattori and D. Satow, Phys. Rev. D , no. 11, 114032(2016).[65] A. Bandyopadhyay, C. A. Islam and M. G. Mustafa,Phys. Rev. D , no. 11, 114034 (2016).[66] S. Ghosh and V. Chandra, arXiv:1808.05176 [hep-ph].[67] C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro,A. Rucci and F. Sanfilippo, Phys. Rev. D , no. 7,074515 (2017).[68] B. Singh, L. Thakur and H. Mishra, Phys. Rev. D97